Comparisons between model equations for long waves (original) (raw)

1991, Journal of Nonlinear Science

Abstract

Considered here are model equations for weakly nonlinear and dispersive long waves, which feature general forms of dispersion and pure power nonlinearity. Two variants of such equations are introduced, one of Korteweg-de Vries type and one of regularized long-wave type. It is proven that solutions of the pure initial-value problem for these two types of model equations are the same, to within the order of accuracy attributable to either, on the long time scale during which nonlinear and dispersive effects may accumulate to make an order-one relative difference to the wave profiles. Key words, long wave models, Korteweg-de Vries-type equations, regularized longwave equations, dispersion relations Earlier Theory and Rationale To understand more precisely the results in view, and to grasp their importance, it is worthwhile to briefly review the theory developed earlier for the special case of the Korteweg-de Vries equation itself in Bona et al. (1983). Among the many assumptions that come to the fore in deriving models like those in (1.1) and (1.2) are that the wave motions in question have small amplitude and large wavelength. Letting a and A, respectively, denote typical, scaled, nondimensionalized values of these quantities, the assumption is that both a and A-l are small. However, in order that nonlinear and dispersive effects be balanced, these two small quantities must be related. In the case of the Korteweg-de Vries equation where p = 1 and re(k) = k 2, so that (1.1) and (1.2) take the forms ~Tt + rlx + rlrlx + rlxxx = 0 (1.4)

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